ARITHMETIC THE SECOND
Kummer / Dedekind
IT is a curious fact that although arithmetic - the theory of
numbers - has been the fertile mother of more profound pro-
blems and powerful methods than any other discipline of
mathematics, it is usually regarded as standing rather to one
side of the mam progress as a more or less cold-blooded spec-
tator of the flashier achievements of geometry and analysis,
particularly in their services to physical science, and compara-
tively few of the great mathematicians of the past 2,000 years
have expended their more serious efforts on the advancement
of the science of 'pure number'.
Many causes have determined this strange neglect of what,
after all, is mathematics par excellence. Among these we need
note only the following: arithmetic at present is on a higher
plane of intrinsic difficulty than the other great fields of mathe-
matics; the immediate applications of the theory of numbers
to science are few and not readily perceptible to the ordinary
run of creative mathematicians, although some o"f the greatest
have felt that the proper mathematics of nature will be found
ultimately in the behaviour of the common whole numbers;
and, finally, it is only human for mathematicians - at least for
some, even the great - to court reputation and popularity in
their own generation by reaping the easier harvests of a spec-
tacular success in analysis, geometry, or applied mathematics.
Even Gauss succumbed, to his keen regret in middle life.
Modern arithmetic - after Gauss - began with Kummer. The
origin of Kummer's theory in his attempt to prove Fermat's
Last Theorem has already been noted (Chapter 25). Something
of the man's long life may be told before we pass to DedekiBd.
Kummer was a typical German of the old school with all the